I’ve been grading a wheelbarrow-load of papers from my upper-level geometry class this morning. It’s been making me think about the jump from taking calculus to courses beyond calculus. A lot of very good calculus students simply hit the wall when they move on to an "upper-level" course, like linear algebra or geometry. The jump is difficult, I think, because there are certain personality traits that have to be in place for a student to succeed past calculus:

You have to become very tough-minded. This means you have to begin to be ruthless in your assessment of your own work and the work of others. If you can do better, you have to develop the urge to do so and not be content with cutting your losses on a problem and moving on. Same goes for the work your classmates are doing.

You have to become self-confident in your mathematical work. In an post-calculus classroom, the correctness of your mathematics is intrinsically, not extrinsically, determined. That means that although there are right and wrong answers out there — and correct and incorrect proofs — the rightness or wrongness is not determined by an authority figure like the back of a book, but rather by the mathematics itself. A proof is correct not because it matches an authority figure’s proof that was published somewhere, but because it meets the standards of logical rigor that a proof requires.

You must learn to obsess over the right details. It’s easy to avoid obsessing at all, or obsessing over trivialities like grades (yes, I mean that). But it’s difficult to ask the right questions and see the right paths in a problem that need to be taken care of.

Others I’m thinking of are harder to enumerate. For example, it’s easy to do well in calculus if you just learn the system and how to work it. Calculus is usually a pretty straightforward course — you do homework, you take tests, etc. Students who get in a comfort zone in a high school calculus coursde get really offended — perhaps scared — when the college-level analog of that course asks them to do more than just calculate derivatives. Likewise, a lot of students decide they want to study mathematics because they figure it’s a system they learned how to play and can continue to play until they get a degree. Usually if you ask, they’ll say they got into math because "it was always easy" and "there’s only one right answer". But pretty quickly after calculus, students become very exposed in their thinking because there is no longer such an emphasis on plug-n-chug calculating.

Additionally, I think a lot of education majors who end up taking these upper-level math courses have a hard time because many of the characteristics of a successful upper-level math student — described by adjectives like tough, demanding, self-confident, meticulous, etc. — sound like the direct opposite of the characteristics of a successful K-12 teacher as advertised by many education schools. The ed schools seem to want teachers who are nurturing, caring, etc. and it’s very hard — certainly not often made clear — that a teacher can, and indeed must, be simultaneously caring and tough, nurturing and demanding, strongly self-confident and open to correction, and so on. The assumption is that the nurturing/caring and the tough/demanding groups of characteristics are mutually exclusive, and the former trumps the latter. It’s a rare ed school that stresses the centrality of the fact that it’s not the teacher’s job to be liked.

2 responses to “Characteristics of upper-level math success”

For me, as with the students you describe, calculus in high school was easy. I was in a comfort zone, not because there weren’t challenging questions, but because the pace was relaxed. (Compared to what would come later.) Once I got to college and found out that instead of being at the top of my class, I was actually somewhere near the bottom, I didn’t have the skills — the study skills, really — to really keep up. So I stumbled through multivariate, and hit the wall at differential equations. That class (18.03) was actually too much with the plug-and-chug, and not enough context. I got lost in memorization, and didn’t take much away from the class. I did better with discrete math (because I loved combinatorics) and okay with algorithms because of the logic. I now wish that I had been tougher, and pushed myself harder, but age 18 is probably the worst time for American teenagers to be faced with major life decisions that require self-discipline.

So I don’t really know what it takes to be a successful upper-level math student, but even to succeed in high school — at least in the calculus class I’m teaching now — my students do have to have the willingness to try something. I think this is perhaps the precursor to being both tough-minded and confident. To stop when you want to throw up your hands and force yourself to try again, to just write your way to a better idea for a solution: that’s what I need to teach them now so they will be ready for you.

Mrc – I’d offer that the precursors to being tough-minded and confident are honesty, courage, and humility. That is — being realistic and fair in one’s assessment of work, not being afraid to try something, and being willing to accept wrongness and correction in stride.

Age 18 is probably jsut as bad for those characteristics as it is for that of self-discipline!